Approximation of holomorphic Legendrian curves with jet-interpolation

Abstract

We prove several interpolation results for holomorphic Legendrian curves lying in an odd dimensional complex Euclidean space with the standard contact structure. In particular, we show that an arbitrary countable set of points in $\mathbb{C}^{2n+1}$ lies on an injectively immersed isotropic surface with a prescribed complex structure. If the set has no accumulation points, the surface may be taken properly embedded. We also prove a Carleman-type theorem for holomorphic Legendrian curves with interpolation. Namely, a Legendrian curve, defined on a certain type of unbounded closed set in a given open Riemann surface $\mathcal{R}$, may be approximated in the $\mathcal{C}^0$-topology by an entire Legendrian curve with prescribed finite-order Taylor polynomials at a closed discrete set of points in $\mathcal{R}$. Under suitable conditions, the approximating map may be made into a proper embedding.